proof of existence of a solution with $ f \in L^1$

Question

Let $\Omega$ be an open domain in $\mathbb {R^n}$ and $f \in L^1(\Omega)$ then how can we prove there is a weak solution $u \in ‎‎ W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ for the problem $$\begin{cases} ‎\Delta^2u=‎‎f & in‎\hspace{.2cm}‎ \Omega \\ u>0 & in ‎\hspace{.2cm}‎ \Omega \\ u=\Delta u =0 & on‎\hspace{.2cm}‎ \partial \Omega ‎ \end{cases}$$‎

Answer 1

This is not correct, for several reasons.

1. You can't expect $u > 0$.
2. If $f \in L^1$, then at best $u \in W^{2,p}$ with $p < \frac{n}{n-2}$. So $u \in W^{2,2}$ if $n \le 3$.
3. If $\Omega$ is an arbitrary open set, then $u$ need not even be in $W^{2,p}$ due to singularities near the boundary.